Boundary conditions changing operators in non conf

arXiv:hep-th/9801089v114Jan1998USC-98-001Boundaryconditionschangingoperatorsinnonconformaltheories.F.LesageandH.Saleur†DepartmentofPhysicsUniversityofSouthernCaliforniaLosAngeles,CA90089-0484Boundaryconditionschangingoperatorshaveplayedanimportantroleinconformalfieldtheory.Here,westudytheirequivalentinthecasewhereamassscaleisintroduced,inanintegrableway,eitherinthebulkorattheboundary.Moreprecisely,weproposeanaxiomaticapproachtodeterminethegeneralscalarproductsbhθ1,...,θm||θ′1,...,θ′niabetweenasymptoticstatesintheHilbertspaceswithaandbboundaryconditionsre-spectively,andcomputethesescalarproductsexplicitelyinthecaseoftheIsingandsinh-Gordonmodelswithamassandaboundaryinteraction.Thesequantitiescanbeusedtostudystatisticalsystemswithinhomogeneousboundaryconditions,and,morein-terestinglymaybe,dynamicalproblemsinquantumimpurityproblems.Asanexample,weobtainaseriesofnewexactresultsforthetransitionprobabilityinthedoublewellproblemofdissipativequantummechanics.01/98†PackardFellow1.IntroductionTheconceptofboundaryconditionschangingoperatorshasbeenofcrucialimportanceintheanalysisofboundaryconformalfieldtheories,aswellasintheirapplicationstoquantumimpurityproblems.Inthelastfewyears,importantprogresshasbeenmadeinextendingthesolutionofconformalfieldtheoriestotheoriesperturbedeitherbyabulkorbyaboundaryoperator.Thelatterhaveextremelyinterestingapplicationstothestudyofflowsinquantumimpurityproblems.Aswewillshowhere,itispossibletointroduceboundaryconditionschangingopera-torseveninthecasewherethereisamassscale,eitherinthebulkorattheboundary.Itisnotcompletelyclearwhattheformaluseoftheseobjects(egasillustratedin[1]fortheconformalcase)mightbe,buttheycertainlydohaveapplications.Froma2D,statisticalmechanics,pointofview,onemightwonderwhatistheeffectofhavingdifferentpartsoftheboundarywithdifferentboundaryconditions[2],describing,forinstanceasituationwitha(classical)“impurity”ontheboundary.Fromthe1+1pointofviewofquantumimpurityproblems,onemightwishtodescribesituationswherethecouplingtotheim-purityischangedatsomeparticulartime,andoneisinterestedinthesubsequenttimeevolutionofthedegreesoffreedom.Infact,oneofthekeyobservablesinthetwostateproblemofdissipativequantummechanics,thewellknownquantity“P(t)”(seebelow),isessentiallydefinedinthatfashion[3].Intheconformalsituation,onlyadiscretesetofconformalboundaryconditionsareavailable,andtheeffectofswitchingfromonetotheotherisdescribedbytheinsertionontheboundaryoftheappropriateoperator.Forinstance,intheIsingmodel,onegoesfromfreetofixedspinsbyinsertingtheconformaloperatorΦ12ofweighth=116,andfromfixeduptofixeddownbyinsertingΦ13ofweighth=12[1](wheretheΦrsaretheusualdegenerateconformalfields[4]).Weareinterestedinthemoregeneralcasewhereconformalinvarianceisbrokenattheboundary,andalsomaybeinthebulk.Asanexample,wecanconsiderasituationwhere,intheIsingmodel,weapplyaboundarymagneticfieldhafory0,hbfory0(weusecoordinatesxandytodescribethetwodimensionalspaceandtheboundarysitsatx=0inthispaper),andinaddition,T6=Tcinthebulk.Ingeneral,onedoesnotexpectthissituationtobedescribedbytheinsertionofasimpleoperatoraty=0anymore.Nevertheless,thechangeofboundaryconditionscanbefullycharacterizedinthefactorizedscatteringdescription(theIsingmodelwithaboundaryfieldbeingintegrable)byscalarproductsofasymptoticstatesintheHilbert1spacewithhaandhb(seebelow),andthe“operator”insertedaty=0canthusbewritten,inprinciple,intermsoftheFaddeevZamolodchikovalgebra.Theproblemthenreducestothedeterminationofscalarproducts(wealsocallthemtransitionfactors),andissimilarinnaturetotheproblemofdeterminingform-factorsinintegrabletheories[5].Inthispaper,wemostlyrestricttotheIsingcase,wherethecomputationsarealreadymoderatelycomplicated,andobtainthecompletesolutiontotheproblemwithchangingboundarymagneticfieldsinthemassivetheory.Insection2,wediscussthegeneralproblemofform-factorsinthecrossedchannel.Wedeterminethescalarproductsofinterestforhahb0insection3.Insection4,wediscussthelimitwherethebulkismassless,includingtheconformalinvariantcase.Insection5,wediscussthecasehahb0whichrequireadifferenttreatment.Section6isdevotedtoextendingsomeoftheseresultstothesinh-Gordonmodel,wherethenontrivialbulkscatteringmatrixintroducesfurthercomplications.Insection7,basedonsomereasonableconjectures,weapplyandgeneralizeourresultstothedeterminationofP(t)indissipativequantummechanics,forarbitraryvalueofthedissipationparameterg.Someoftheresultspresentedinthispaperhaveappearedinashorterversion[6].2.FormfactorsinthecrossedchannelConsiderthemassiveIsingmodeldefinedinthehalfplanex∈(−∞,0],y∈(−∞,∞),inthepresenceofaboundarymagneticfield.TheactionreadsA=Z0−∞dxZ∞−∞dyaFF(x,y)+12Z∞−∞dyψ¯ψ
(x=0)+a˙a+hZ∞−∞dyσB(y).(2.1)HereaFFistheusualmassivefreeMajoranafermionaction,aisaboundaryfermionsatisfyinga2=1,σBistheboundaryspinoperator,whichcoincideswith12ψ+¯ψ
a.Asdiscussedin[7]theproblemcanbestudiedfromthepointofviewofthedirectchannelwithxtakenastimeandyasspace.InthatcasetheHilbertspaceistheusualone,andtheboundaryrepresentedbyaboundarystate.Ontheotherhand,inthecrossedchannel,onehasanewHilbertspaceforthetheoryonth